On the set $\mathbb{Q}$ of rational numbers, the 3 -adic metric $d_{3}$ is defined as follows: for $x, y \in \mathbb{Q}$, define $d_{3}(x, x)=0$ and $d_{3}(x, y)=3^{-n}$, where $n$ is the integer satisfying $x-y=3^{n} u$ where $u$ is a rational number whose denominator and numerator are both prime to 3 .

(1) Show that this is indeed a metric on $\mathbb{Q}$.

(2) Show that in $\left(\mathbb{Q}, d_{3}\right)$, we have $3^{n} \rightarrow 0$ as $n \rightarrow \infty$ while $3^{-n} \nrightarrow \infty$ as $n \rightarrow \infty$. Let $d$ be the usual metric $d(x, y)=|x-y|$ on $\mathbb{Q}$. Show that neither the identity map $(\mathbb{Q}, d) \rightarrow\left(\mathbb{Q}, d_{3}\right)$ nor its inverse is continuous.